1. In a large deep skillet, melt the butter in 1 tablespoon of the olive oil. Add the fennel, cover and cook over moderate heat until the fennel is wilted, about 6 minutes. Uncover and cook, stirring occasionally, until very soft and just caramelized, about 12 minutes. Season with salt and pepper and let cool slightly.
2. In a small bowl, combine the bread crumbs with the remaining 2 tablespoons oil. Season lightly with salt and pepper; set aside.
3. Preheat the oven to 350° and butter a 13-by-9-inch baking dish. Spread one-third of the potatoes in an even layer in the dish. Season lightly with salt and pepper and top with one-third of the fennel. Repeat the layering with the remaining potatoes and fennel, seasoning the layers and ending with the fennel.
4. In a large measuring cup, combine the cream and stock and pour over the vegetables, rearranging the fennel if it's dislodged by the liquid. Cover the dish tightly with foil and bake for about 1 hour, or until the potatoes are just tender; set a baking sheet underneath the dish to catch any drips. Remove the foil, sprinkle the top with the bread crumbs and bake for about 30 minutes longer, or until the top is golden brown and the potatoes are very tender. Let the gratin rest for 10 minutes before cutting.
Make Ahead: The gratin can be refrigerated overnight. Rewarm, covered, in a 350° oven before serving.